Multicarrier receiver for multiple-input multiple-output wireless communication systems and method

ABSTRACT

Embodiments of a multicarrier receiver and method for generating soft bits in a multiple-input multiple-output system are generally described herein. In some embodiments, operational parameters for an equalizer and a soft-bit demapper in a multicarrier receiver are determined. Other embodiments may be described and claimed.

TECHNICAL FIELD

Embodiments of the present invention pertain to wireless communicationsystems. Some embodiments pertain to multicarrier receivers. Someembodiments pertain to receivers that receive signals through more thanone antenna.

BACKGROUND

Multiple-input-multiple-output (MIMO) wireless communication systemsconcurrently transmit information using more than one spatial channel.Some of these communication systems use multiple subcarriers (i.e.,tones) and modulate subsymbols on the subcarriers. These systems maytransmit simultaneously using multiple antennas with each antennatransmitting using the same subcarriers.

During propagation to the receiver, the subsymbols carried on eachsubcarrier and spatial channel are generally distorted by backgroundnoise and cross-talk, among other things. Cross-talk may refer to thedistorting influence that a particular subsymbol transmitted on aparticular subcarrier and spatial channel has upon another subsymbolcarried on the same subcarrier of another spatial channel. Separatingthe transmitted spatial data streams and removing the effects ofcross-talk and noise at the receiver is a complex process and is veryprocessing intensive.

Thus there are general needs for separating spatial data streams andremoving the effects of cross-talk and noise at the receiver withreduced complexity and a reduced number of operations.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a multicarrier receiver in accordance withsome embodiments of the present invention;

FIG. 2 is a block diagram of processing circuitry for generating softbits in accordance with some embodiments of the present invention;

FIG. 3 is a block diagram of processing circuitry for generating softbits in accordance with some other embodiments of the present invention;

FIG. 4 is a block diagram of processing circuitry for generating softbits in accordance with some other embodiments of the present invention;

FIG. 5 illustrates a training unit in accordance with some embodimentsof the present invention;

FIG. 6 illustrates a training unit in accordance with some otherembodiments of the present invention;

FIG. 7 illustrates a training unit in accordance with some otherembodiments of the present invention; and

FIG. 8 is a flow chart of a matrix processing procedure in accordancewith some embodiments of the present invention.

DETAILED DESCRIPTION

The following description and the drawings illustrate specificembodiments of the invention sufficiently to enable those skilled in theart to practice them. Other embodiments may incorporate structural,logical, electrical, process, and other changes. Examples merely typifypossible variations. Individual components and functions are optionalunless explicitly required, and the sequence of operations may vary.Portions and features of some embodiments may be included in orsubstituted for those of others. Embodiments of the invention set forthin the claims encompass all available equivalents of those claims.Embodiments of the invention may be referred to, individually orcollectively, herein by the term “invention” merely for convenience andwithout intending to limit the scope of this application to any singleinvention or inventive concept if more than one is in fact disclosed.

FIG. 1 is a block diagram of a multicarrier receiver in accordance withsome embodiments of the present invention. Multicarrier receiver 150receives multicarrier signals through two or more antennas 152 andgenerates decoded bit stream 165 by processing the received signals.Multicarrier receiver 150 includes Fourier transform (FT) circuitry 154,equalizer 156, training unit 158, soft-bit demappers 160,space-frequency de-interleaver 162 and decoder 164.

Multicarrier receiver 150 may include as many or more receive antennas152 as was used by a transmitting station (not shown). Each antenna 152may be coupled (indirectly) to Fourier transform circuitry 154, whichmay perform fast Fourier transforms. Each of Fourier transform circuitry154 may provide frequency domain signals 155 comprising a set of complexnumbers that may describe the frequency content received on each tone.For example, if a number of subcarriers is employed by multicarrierreceiver 150, then that number of complex numbers may be provided byeach Fourier transform circuitry 154.

Each Fourier transform circuitry 154 may be coupled to equalizer 154.When receiver 150 employs R antennas 152, then equalizer 156 may receivea quantity of R times the number of subcarriers complex numbers persymbol from Fourier transform circuitry 154. The quantity of antennas Rmay range from two to up to ten or more. Equalizer 156 may substantiallyremove the effects of cross-talk and noise and may compute an“equalized” set of complex numbers. One complex number may be used todescribe the frequency content for each tone and spatial channelemployed by the transmitting station.

Receiver 150 may also include multiple soft-bit demappers 160 coupled toequalizer 156. One demapper 160 may be provided for each spatial channelemployed by the transmitting station. Each demapper 160 may generate aset of soft bits for each of the subcarriers. Each soft bit of asubsymbol may represent the likelihood that a given subsymbol on a giventone corresponds to a particular soft bit being a one or a zero. Softbits 119 from demappers 160 may be provided to space-frequencyde-interleaver 162 for deinterleaving and decommutation to generate oneor more streams of soft bits which may be provided to decoder 164.Decoder 164 may recover hard bits (i.e., information) from the soft bitsand may generate decoded bit stream 165.

In some embodiments, equalizer 156 and soft-bit demappers 160 aresupplied with parameters. In a packet-based system, data is transmittedin the form of a “packet,” which includes a preamble followed by data.During processing of the preamble of each packet, the parameters forequalizer 156 and soft-bit demappers 160 may be calculated. The processof calculating the parameters and providing the parameters to demappers160 and equalizer 156 is referred to as training, which may be performedby training unit 158.

In some embodiments, training unit 158 may calculate the parameters in amanner involving a reduced or a minimal complexity. Complexity may bemeasured by the number of real or complex multiplication and divisionoperations. In some embodiments, training unit 158 may calculate theaforementioned parameters using a reduced number of division andmultiplication operations, although the scope of the invention is notlimited in this respect. For similar reasons, soft-bit demappers 160 andequalizer 156 may operate in a manner involving reduced complexity,although the scope of the invention is not limited in this respect.

In some embodiments, space-frequency de-interleaver 162 may include oneor more deinterleavers 172 and spatial de-commutator 174. Deinterleavers172 may perform deinterleaving operations on the soft bits generatedfrom soft-bit demappers 160 for each spatial channel and spatialde-commutator 174 may combine the deinterleaved soft bits generated fromdeinterleavers 172. Although space-frequency de-interleaver 162 isillustrated has having deinterleavers 172 and spatial de-commutator 174,this is not a requirement as space-frequency de-interleaver 162 mayinclude other functional elements and configurations to deinterleave andcombine soft bits for decoder 164.

In some embodiments, equalizer 156 may multiply each tone of frequencydomain signals 155 generated from each of a plurality of receive signalpaths by equalizer coefficient matrix 161 to remove at least some ormost crosstalk (e.g., on the same tone) and noise to generate spatiallyequalized frequency domain signals 157 for each subcarrier. In theseembodiments, soft-bit demapper 160 for each spatial channel may multiplythe spatially equalized frequency domain signals 157 by soft-bit weights159 for use in generating soft bits 119. In some embodiments, trainingunit 158 may generate equalizer coefficient matrix 161 for eachsubcarrier and may generate soft-bit weights 159 from effective channeltaps. In some embodiments, training unit 158 may generate equalizercoefficient matrix 161 for each subcarrier and may generate soft-bitweights 159 from a channel estimate matrix, an average energy oftransmitted sub-symbols, and/or a matrix related to a noise correlationmatrix. These embodiments are discussed in more detail below.

In some embodiments, equalizer 156, soft-bit demappers 160 and trainingunit 158 may be part of the baseband processing circuitry of a wirelesscommunication device. In some embodiments, multicarrier receiver 150 maybe part of a wireless communication device that may transmit orthogonalfrequency division multiplexed (OFDM) communication signals over amulticarrier communication channel. The multicarrier communicationchannel may be within a predetermined frequency spectrum and maycomprise a plurality of orthogonal subcarriers. In some embodiments, theorthogonal subcarriers may be closely spaced OFDM subcarriers. To helpachieve orthogonality between the closely spaced subcarriers, eachsubcarrier may have a null at substantially a center frequency of theother subcarriers and/or each subcarrier may have an integer number ofcycles within a symbol period, although the scope of the invention isnot limited in this respect. In some embodiments, the number ofsubcarriers may range from as few as 48 or less to as great as 256 ormore.

In some embodiments, multicarrier receiver 150 may be part of a wirelessaccess point (AP), such as a Wireless Fidelity (WiFi), WorldwideInteroperability for Microwave Access (WiMax), or broadbandcommunication station, although the scope of the invention is notlimited in this respect as multicarrier receiver 150 may be almost anywireless communication device. In some embodiments, multicarrierreceiver 150 may be part of a portable wireless communication device,such as personal digital assistant (PDA), a laptop or portable computerwith wireless communication capability, a web tablet, a wirelesstelephone, a wireless headset, a pager, an instant messaging device, adigital camera, an access point, a television or other device that mayreceive and/or transmit information wirelessly. In some broadband andWiMax embodiments, multicarrier receiver 150 may be part of a receivingstation.

In some embodiments, the frequency spectrums for a multicarriercommunication signal may comprise either a 5 gigahertz (GHz) frequencyspectrum or a 2.4 GHz frequency spectrum. In these embodiments, the 5GHz frequency spectrum may include frequencies ranging fromapproximately 4.9 to 5.9 GHz, and the 2.4 GHz spectrum may includefrequencies ranging from approximately 2.3 to 2.5 GHz, although thescope of the invention is not limited in this respect, as otherfrequency spectrums are also equally suitable. In some broadband andWiMax embodiments, the frequency spectrum for communications maycomprise frequencies between 2 and 11 GHz, although the scope of theinvention is not limited in this respect.

In some embodiments, multicarrier receiver 150 may communicate inaccordance with specific communication standards, such as the Instituteof Electrical and Electronics Engineers (IEEE) standards including IEEE802.11 (a), 802.11 (b), 802.11 (g), 802.11 (h) and/or 802.11 (n)standards for wireless local area networks (WLANs), althoughmulticarrier receiver 150 may also be suitable to receive communicationsin accordance with other techniques including the Digital VideoBroadcasting Terrestrial (DVB-T) broadcasting standard, and the Highperformance radio Local Area Network (HiperLAN) standard. In somebroadband and WiMax embodiments, multicarrier receiver 150 may receivebroadband wireless communications in accordance with the IEEE 802.16(e)standards for wireless metropolitan area networks (WMANs). For moreinformation with respect to IEEE 802.11 standards, please refer to “IEEEStandards for Information Technology—Telecommunications and InformationExchange between Systems—Local and Metropolitan Area Network—SpecificRequirements—Part 11: Wireless LAN Medium Access Control (MAC) andPhysical Layer (PHY), ISO/IEC 8802-11: 1999” and relatedamendments/versions.

Antennas 152 may comprise one or more directional or omnidirectionalantennas, including, for example, dipole antennas, monopole antennas,patch antennas, loop antennas, microstrip antennas or other types ofantennas suitable for transmission of RF signals. In some embodiments,instead of two or more antennas, a signal antenna with multipleapertures may be used.

Although multicarrier receiver 150 is illustrated as having severalseparate functional elements, one or more of the functional elements maybe combined and may be implemented by combinations ofsoftware-configured elements, such as processing elements includingdigital signal processors (DSPs), and/or other hardware elements. Forexample, some elements may comprise one or more microprocessors, DSPs,application specific integrated circuits (ASICs), and combinations ofvarious hardware and logic circuitry for performing at least thefunctions described herein. In some embodiments, the functional elementsof the processing circuitry may refer to one or more processes operatingon one or more processing elements.

FIG. 2 is a block diagram of processing circuitry for generating softbits in accordance with some embodiments of the present invention.Processing circuitry 200 generates soft bits 119 from frequency domainsignals 201. Processing circuitry 200 includes training unit 230,equalizer 202 and soft-bit demapper 204. In some embodiments, equalizer202 may correspond to equalizer 156 (FIG. 1), soft-bit demapper 204 maycorrespond to one of soft-bit demappers 160 (FIG. 1), and training unit230 may correspond to training unit 158 (FIG. 1), although the scope ofthe invention is not limited in this respect.

In accordance with some embodiments, training unit 230 may provideoperational parameters to equalizer 202 and soft-bit demapper 204. Insome embodiments, equalizer 202 may be a minimum mean squared error(MMSE) equalizer, although the scope of the invention is not limited inthis respect. In some embodiments, the parameters provided to equalizer202 and soft-bit demapper 204 include W(k), λ_(t)(k), and λ_(t)(k)-1,which is also referred to herein as μ_(t)(k).

In some embodiments, training unit 230 may include four computationalunits 206, 208, 210, and 212. Computational units 206, 208, 210, and 212may include one or more circuits for multiplying, dividing, adding, andsubtracting, in order to carry out the calculations described below.Computational units 206, 208, 210, and 212 may also include look-uptables, as may be useful in the case of performing reciprocation, forexample. Computational units 206, 208, 210, and 212 may also be embodiedas software or firmware code stored in a medium (such as a memorydevice) in some embodiments.

In some embodiments, first computational unit 206 may compute twomatrices, A(k) and D(k), based upon three inputs, σ_(x) ², H(k), andR_(nn)(k). The variable k represents the tone index. Thus, many of thevalues depicted and/or discussed herein may vary from tone to tone, andmay be calculated individually for each tone employed by thecommunication system, although the scope of the invention is not limitedin this respect. For example, W(k) and λ_(t)) may vary from tone totone, and may be calculated separately for each tone employed by thecommunication system. The discussion herein focuses upon the operationof training unit 230, equalizer 202, and soft-bit demapper 204, as itpertains to a single tone.

In some embodiments, σ_(x) ² represents the average energy of thetransmitted subsymbols 115. H(k)represents channel estimate 105, asobserved at every tone and receiver antenna. The number of receivesignal paths or receive antennas is denoted by R, and the number oftransmitted spatial data streams or transmission antennas is denoted byT. HWk) may be an R×T matrix. R_(nn)(k) may be noise correlation matrix125, which is of dimension R×R. σ_(x) ², H(k), and R_(nn)(k) are inputsthat may be provided to training unit 230 with the reception of eachpacket. Their calculation is well understood by those of ordinary skillin the art of receiver design, and is not discussed herein.

First computational unit 206 may operate by initially calculating D(k),according to the following formula:D(k)=R _(nn) k)/σ_(x) ².

Optionally, training unit 230 may be designed to assign σ_(x) ² a valueof one. Such a design choice reduces complexity, and merely alters thevalues found in the channel estimate matrix, H(k).

After calculating of D(k), A(k) may be calculated as a function of D(k),based upon the following formula:A(k)=H(k)H ^(†)(k)+D(k),

where H^(†)(k) represents the conjugate transpose matrix of H(k). Aftercalculation of A(k), this value may be provided to second computationalunit 208. Second computational unit 208 may use the A(k) matrix tocalculate equalizer coefficient matrix W(k) which may be based upon thefollowing formula:W(k)=H ^(†)(k)A ⁻¹(k),

where A⁻¹(k) represents the inverse matrix of A(k).

Second computational unit 208 may provide equalizer coefficient matrixW(k) to equalizer 202. Upon reception of equalizer coefficient matrixW(k), equalizer 202 may be said to be trained or initialized. Equalizer202 may include matrix multiplication unit 114, which may include one ormore circuits for multiplying, adding, and subtracting, in order tocarry out the computations described below. Matrix multiplication unit114 may also be embodied as software or firmware code stored in a medium(such as a memory device) in some embodiments. During operation,equalizer 202 may pre-multiply y(k) by W(k), yielding z(k), where y(k)is an R-dimensional vector containing complex numbers representingfrequency content observed on a given tone, k, at each of the Rreception antennas. y(k) may be provided to equalizer 202 by a set of RFT units. The values in y(k) may be distorted by cross-talk andbackground noise, among other things.

Multiplying y(k) by W(k) to generate z(k) may produce a T-dimensionalvector, representing, for a given tone, k, the frequency content thatequalizer 202 calculates was emitted from each of the T transmissionantennas. For example, z₁(k) represents the frequency content thatequalizer 202 calculates was emitted on tone k from the firsttransmission antenna (or the first spatial channel). Similarly, z₂(k)represents the frequency content that equalizer 202 calculates wasemitted on tone k from the second transmission antenna (or the secondspatial channel), and so on. Thus, multiplication of y(k) by theequalizer coefficient matrix, W(k), approximately removes the effects ofcross-talk and noise.

After calculation of z(k), the values from this vector may be providedto each of T soft-bit demappers 204 (only one of which is shown in FIG.2). The operation of soft-bit demapper 204 is discussed in greaterdetail, below.

In some embodiments, equalizer coefficient matrix W(k) is also providedto third computational unit 210. Third computational unit 210 may yielda T-dimensional vector that includes each of T effective channel taps211, shown as e_(tt)(k). The vector yielded from third computationalunit 210 may be of the following nature: <e₁₁(k), e₂₂(k), . . .e_(TT)(k)>, where e_(tt)(k) represents an entry on the diagonal of theeffective channel matrix, E(k), which may be calculated as:E(k)=W(k)H(k)

For example, e₁₁(k) may represent the element in the first row and firstcolumn of E(k), and e₂₂(k) may represent the element in the second rowand second column therein. Notably, the output of third computationalunit 210 requires only the diagonal elements of the effective channelmatrix E(k) to be calculated (e₁₁(k), e₂₂(k), etc.), as opposed tocalculating each element in E(k). Conventional methods of training asoft-bit demapper may require computation of every element in E(k).Since E(k) is a T×T matrix, the conventional method requires thecalculation of T² elements. On the other hand, in accordance with someembodiments, training soft-bit demapper 204 may require only thecalculation of the diagonal elements of E(k). In some embodiments, onlyT elements need to be calculated. Consequently, the off-diagonalelements of the effective channel matrix E(k) do not need to becalculated. This may result in a significant reduction in the complexityof training unit 230.

In some embodiments, each of the effective channel taps e_(tt)(k) 211may be provided to fourth computational unit 212, which may computeoutput 213. Output 213 may be equal to 1/(1−e_(tt)(k)), for eacheffective channel tap provided thereto. The value yielded from fourthcomputational circuit 212 is termed λ_(t)(k). In some embodiments,fourth computational circuit 212 may compute output 213 using thefollowing equation:λ_(t)(k)=1/(1−e _(tt)(k)).

In some embodiments, fourth computational unit 212 may be areciprocation circuit or a look-up table, among other things. In someembodiments, output 213 from fourth computational unit 212 may beprovided to each soft-bit demapper 204, only one of which is depicted inFIG. 2. For example, λ₁(k) may be provided to a soft-bit demappercorresponding to the first spatial channel, λ₂(k) may be provided to asoft-bit demapper corresponding to the second spatial channel, and soon. The value from fourth computational unit 212 may be provided tomultiplication unit 116, which may be an element-wise multiplicationunit and may be part of soft-bit demapper 204, although the scope of theinvention is not limited in this respect.

In some embodiments, the element-wise multiplication performed bymultiplication unit 116 may multiply the t-th element of z(k) byλ_(t)(k). In other words, z₁(k) is multiplied by λ₁(k) in a soft-bitdemapper corresponding to the first spatial channel. Similarly, z₂(k) ismultiplied by λ₂(k) in a soft-bit demapper corresponding to a secondspatial channel, and so on. Thus, λ_(t)(k) may be applied as a soft-bitweight, which is a function of spatial channel and tone. The resultingproduct is termed {tilde over (z)}_(t)(k), which is provided to soft-bitgenerator 218, the operation of which is discussed below.

The output of fourth computational unit 212 may also be provided toadder unit 120, yielding output 121 of λ_(t)(k)−1, which is termedμ_(t)(k). Soft-bit generator 218 may calculate a set of soft bits for agiven tone, k, and spatial channel, t, based upon {tilde over(z)}_(t)(k) and μ_(t)(k). For example, assuming that the transmissionsystem employed a modulation scheme of 16-QAM (quadrature amplitudemodulation), then soft-bit generator 218 may yield four soft bits whichmay be calculated as follows:Soft bit₁=Re{{tilde over (z)}_(t)(k)}Soft bit₂=(2σ_(x)/10^(1/2))*μ_(t)(k)−abs(soft bit₁)Soft bit₃=Im{{tilde over (z)}_(t)(k)}Soft bit₄=(2σ_(x)/10^(1/2))*μ_(t)(k)−abs(soft bit₃)

The number of soft bits 119 generated by soft-bit generator 218 maydepend on the modulation level used by the transmitting station. Forexample, when binary phase shift keying (BPSK) is employed, one soft bitper symbol is generated, when quadrature phase shift keying (QPSK) isemployed, two soft bits per symbol are generated, when 8PSK is employed,three soft bits per symbol are generated, when 16-quadrature amplitudemodulation (16-QAM) is employed, four soft bits per symbol aregenerated, when 32-QAM is employed, five soft bits per symbol aregenerated, when 64-QAM is employed, six soft bits per symbol aregenerated, when 128-QAM is employed, seven soft bits per symbol aregenerated, and when 256-QAM is employed, eight soft bits per symbol aregenerated. Soft-bit generator 218 may generate a greater number of softbits per symbol when higher data communication rates per subcarrier areemployed by the transmitter. Irrespective of the modulation techniqueemployed by the communication system, soft-bit generator 218 generatessoft bits 119 based on {tilde over (z)}_(t)(k) and μ_(t)(k).

The techniques described above for generating the operational parametersfor equalizer 202 and demapper 204 may result in a reduction ofcomplexity, when compared to more conventional techniques. For example,in a conventional three-transmission antenna and four-reception antennasystem, training may require approximately: (1) 265 complexmultiplications and 7 real division operations, assuming spatiallycorrelated receiver noise; (2) 229 complex multiplications and 7 realdivision operations, assuming spatially uncorrelated noise, unequal fromreceiver antenna to receiver antenna; and (3) 220 complexmultiplications and 7 real division operations, assuming spatiallyuncorrelated noise, equal from receiver antenna to receiver antenna. Inaccordance with some embodiments of the present invention, training unit230 may train equalizer 202 and demapper 204 in approximately 172complex multiplications and 7 real divisions, regardless of whether thenoise is correlated or uncorrelated. Furthermore, (again in the contextof a three-transmission antenna and four-reception antenna system) aconventional equalizer and demapper may require approximately 33 complexmultiplications to perform their above-recited operations. In accordancewith some embodiments of the present invention, processing circuitry200, on the other hand, may require only approximately 15 complexmultiplications.

FIG. 3 is a block diagram of processing circuitry for generating softbits in accordance with some other embodiments of the present invention.Processing circuitry 300 includes training unit 330, equalizer 302 andsoft-bit demapper 304. In some embodiments, equalizer 302 may correspondto equalizer 156 (FIG. 1), soft-bit demapper 304 may correspond to oneof soft-bit demappers 160 (FIG. 1), and training unit 330 may correspondto training unit 158 (FIG. 1), although the scope of the invention isnot limited in this respect. Training unit 330 provides operationalparameters to equalizer 302 and soft-bit demapper 304. The parametersprovided to equalizer 302 and soft-bit demapper 304 may include {tildeover (W)}(k) and λ_(t)(k)−1, which may also be referred to herein asμ_(t)(k). Based upon these parameters, equalizer 302 and demapper 304may carry out the functions described above, and described in greaterdetail herein.

As illustrated in FIG. 3, soft-bit demapper 304 does not include anelement-wise multiplication unit for application of soft-bit weights 213(FIG. 2) (as was the case in FIG. 2, for example). Instead, equalizationand application of soft-bit weights may be accomplished viamultiplication unit 312 in equalizer 302. Sometimes, a design choice toreduce operational complexity by combining multiple multiplication stepsinto a single step increases training complexity. In these embodiments,the operational complexity and training complexity are both reduced byperforming such a combination.

In these embodiments, training unit 330 includes three computationalunits 306, 308, and 310. Computational units 306, 308, and 310 mayinclude one or more circuits for multiplying, dividing, adding, andsubtracting, in order to carry out the calculations described below.Computational units 306, 308, and 310 may also include look-up tables,as may be useful in the case of performing reciprocation, for example.Computational units 306, 308, and 310 may also be embodied as softwareor firmware code stored in a medium (such as a memory device) in someembodiments.

First computational unit 306 calculates two matrices, B(k) and C(k),based upon three inputs, σ_(x) ², H(k), and D(k). Thus, many of thevalues depicted and/or discussed herein may vary from tone to tone, andare calculated individually for each tone employed by the communicationsystem. For example, {tilde over (W)}(k) and μ_(t)(k) vary from tone totone, and may be calculated separately for each tone employed by thecommunication system. The discussion herein focuses upon the operationof training unit 330, equalizer 302, and soft-bit demapper 304, as itpertains to a single tone. The inputs to first computational units 306may be defined as follows. σ_(x) ² represents the average energy of thetransmitted subsymbols. H(k) represents the channel estimate, asobserved at every tone and receiver antenna. The total number ofreceiver antennas may be denoted by the variable R, and the total numberof transmission antennas used by a transmitting station is denoted bythe variable T. H(k) is an R×T matrix. Finally, D(k) is defined asR_(nn)(k)/σ_(x) ^(2,) where R_(nn)(k) is the noise correlation matrix,which is of dimension R×R. Training unit 330 may be designed to assignσ_(x) ² a value of one. Such a design choice reduces complexity, forexample, by simplifying D(k) to equal R_(nn)(k), thereby eliminating adivision operation, and may alter the values found in the channelestimate matrix, H(k).

First computational unit 306 may operate by initially calculating C(k),according to the following formula:C(k)=H ^(†)(k)D ⁻¹(k)

Notably, the process of finding the inverse of D(k) may be simplified,depending upon the statistical characteristics of the noise. If thenoise is uncorrelated, then D(k) is a diagonal matrix (i.e., itsnon-diagonal elements equal zero), meaning that its inverse can be foundby reciprocating the diagonal elements. In FIG. 3, complexity reductionis achieved whether or not the noise is correlated, but is particularlysuited to application in an environment in which noise is uncorrelatedbetween the various R antennas.

By finding the inverse matrix of D(k), the training parameter {tildeover (W)}(k) may be arrived at with the necessity of finding of only oneother inverse matrix. As will be seen below, the additional inversematrix to be found is of the dimension T×T, in comparison to theoperations performed by training unit 230 (FIG. 2), for example, whichshows that an inverse of an R×R matrix (i.e., A(k)) may be found in theprocess of calculating W(k). When R≧T, the process performed by trainingunit 230 (FIG. 2) may require the finding of an inverse matrix that isof a dimension greater than the process performed by training unit 330.This may help reduce the complexity of processing circuitry 300 ascompared to processing circuitry 200 (FIG. 2).

C(k) may be supplied to third computational unit 310, the operation ofwhich is discussed below, and first computational unit 306 calculatesB(k) according to the following formula:B(k)=C(k)H(k)+I _(T),

where I_(T) represents an identity matrix of dimension T×T. Theresulting matrix, B(k) is also a T×T matrix. B(k) is supplied to secondcomputational unit 308.

Second computational unit 308 performs the task of finding the inverseof B(k), which is referred to as G(k). Since B(k) is a T×T matrix, areduction in complexity is realized in second computational unit 308, asdescribed above. G(k) is supplied to third computational unit 310.

The reciprocal of the diagonal elements in G(k) yield the λ_(t)(k)values described with reference to FIG. 2. In other words,1/g₁₁(k)=λ₁(k), 1/g₂₂(k)=λ₂(k), and so on. Second computational unit 308constructs a T×T matrix, Λ(k), by inserting λ_(t)(k) in its diagonalpositions (e.g., λ₁(k) is inserted in the first row and first column,λ₂(k) is inserted in the second row and second column, and so on), whileall non-diagonal elements are assigned a value of zero. Secondcomputational unit 308 also supplies Λ(k) to the third computationalunit.

Third computational unit 310 multiplies its three inputs, to yield{tilde over (W)}(k), based on the following equation:{tilde over (W)}(k)=Λ(k)G(k)C(k)

{tilde over (W)}(k) is then supplied by third computational unit 310 toequalizer 302. Upon having received {tilde over (W)}(k), equalizer 302is said to have been trained or initialized. Equalizer 302, may be aMMSE equalizer and may include matrix multiplication unit 312. Matrixmultiplication unit 312 may include one or more circuits formultiplying, adding, and subtracting, in order to carry out thecalculations described below. Matrix multiplication unit 312 may also beembodied as software or firmware code stored in a medium (such as amemory device) in some embodiments. During operation, equalizer 302 maymultiply y(k) by {tilde over (W)}(k), yielding {umlaut over (z)}(k),where y(k) is an R-dimensional vector containing complex numbersrepresenting frequency content observed on a given tone, k, at each ofthe R reception antennas. y(k) is provided to equalizer 302 by a set ofR Fourier transform circuitry 154 (FIG. 1), for example. The values ofy(k) may be distorted by the dual forces of cross-talk and backgroundnoise.

Since W(k)=G(k)C(k), it can be seen that the effect of matrixmultiplication unit 312 is to combine the matrix multiplication processof equalizer 202 (FIG. 2) with the element-wise multiplication of thesoft-bit weights by multiplier 116 (FIG. 2) into a single matrixmultiplication operation carried out by matrix multiplication unit 312.Thus, the output of matrix multiplication unit 312, {tilde over (z)}(k),is a T-dimensional vector that may be identical to that emanating fromelement-wise multiplication unit 116 (FIG. 2).

As mentioned above, each of the λ_(t)(k) values generated by secondcomputational unit 308 is provided to adder unit 314, yielding an outputof λ_(t)(k)−1, which is termed μ_(t)(k). Soft-bit demapper 304 maycalculate a set of soft bits for a given tone, k, and spatial channel,t, based upon {tilde over (z)}_(t)(k) and μ_(t)(k). For example,assuming that the transmission system employed a modulation scheme of16-QAM, then soft-bit demapper 304 yields four soft bits that may becalculated as described with reference to FIG. 2. As was the case withrespect to processing circuitry 200 (FIG. 2), irrespective of themodulation technique employed by the communication system, soft-bitdemapper 304 may generate soft bits having knowledge of {tilde over(z)}_(t)(k) and μ_(t)(k).

A reduction in complexity may result from the implementation ofprocessing circuitry 300. For example, in the context of athree-transmission antenna and four-reception antenna system, trainingunit 230 (FIG. 2) can train equalizer 202 (FIG. 2) and demapper 204(FIG. 2) in approximately 172 complex multiplications and 7 realdivisions, regardless of whether the noise is correlated oruncorrelated. However, training unit 330 may be able to train equalizer302 and demapper 304 in approximately 141 complex multiplications and 6divisions, assuming correlated noise, and in 105 multiplications and 6divisions, assuming uncorrelated noise. Furthermore, (again in thecontext of a three-transmission antenna and four-reception antennasystem) equalizer 202 (FIG. 2) and demapper 204 (FIG. 2) may require atotal of approximately 15 complex multiplications to perform theirabove-recited operations. Processing circuitry 300 may require onlyabout 12 complex multiplications.

In some embodiments, the training units of FIGS. 2 and 3 may beconfigured in accordance with the following principle: the equalizercoefficient matrix, W(k), may be scaled by any value s, withoutaffecting performance, as long as the scale factor s is common to allspatial streams, although the scope of the invention is not limited inthis respect. In some embodiments, the scale factor s may vary with thetone index k, which would be designated as s(k), however for notationalsimplicity, the tone index k will be suppressed herein.

The significance of being able to scale W(k) by a factor, s, is thatW(k) may be calculated without the performance of a division operation.In other words, where a division operation would otherwise be requiredto find a particular element in the equalizer coefficient matrix, W(k),the division may be eliminated, and the divisor may be used as a factorto determine s. In other words, s=d₁ * d₂ * d₃ . . . , where d₁, d₂, andd₃ represent would-be divisors in foregone division operations. Thescale factor s may equal the least common multiple of d₁, d₂, and d₃, ormay equal any multiple thereof.

When the equalizer coefficient matrix W(k) is scaled by a factor s, thetwo other operational parameters are affected as follows:λ_(s)=1/(s−e _(s))μ_(s) =sλ _(s)−1,

where the subscript “s” denotes the “scaled version” of the particularvariable. For example, λ_(s) refers to the version of λ_(t)(k) that maybe used when W_(s)(k) is used by the training units of FIGS. 2 and 3,instead of W(k).

Since λ_(s) and μ_(s) are calculated differently than λ_(t)(k) andμ_(t)(k), the training units of FIGS. 2 and 3 may be modified toimplement the above-mentioned formulas for λ_(s) and μ_(s), if a scaledversion of W(k) is used (i.e., if W_(s)(k) is used).

FIG. 4 is a block diagram of processing circuitry for generating softbits in accordance with some other embodiments of the present invention.Processing circuitry 400 includes training unit 430, equalizer 412 andsoft-bit demapper 414. In some embodiments, equalizer 412 may correspondto equalizer 156 (FIG. 1), soft-bit demapper 414 may correspond to oneof soft-bit demappers 160 (FIG. 1), and training unit 430 may correspondto training unit 158 (FIG. 1), although the scope of the invention isnot limited in this respect. Processing circuitry 400 is modified fromprocessing circuitry 200 (FIG. 2) to use W_(s)(k), instead of W(k). Insome embodiments, processing circuitry 400 may differ from processingcircuitry 200 (FIG. 2) in three ways.

First, equalizer calculation unit 408 is configured to produce a scaledversion of W(k), W_(s)(k). W_(s)(k) is calculated without a divisionoperation, as discussed above. A second difference between processingcircuitry 400 and processing circuitry 200 (FIG. 2) is found in fourthcomputational unit 402. Fourth computational unit 402 is configured toyield 1/(s−input), as opposed to 1 /(1−input) as in fourth computationalunit 212 (FIG. 2).

Finally, training unit 430 includes multiplication unit 404 interposedbetween fourth computational unit 402 and addition unit 406.Multiplication unit 404 multiplies the output of fourth computationalunit 402 by the scaling factor, s, prior to its delivery to additionunit 406.

Training unit 430 exhibits a reduction in training complexity by using ascaled equalizer coefficient matrix, W_(s)(k). For example, in thecontext of a three-transmission antenna and four-reception antennasystem, training unit 430 can accomplish training with the use ofapproximately 96 complex multiplications and approximately 3 divisionoperations, assuming uncorrelated noise.

FIG. 5 illustrates a training unit in accordance with some embodimentsof the present invention. Training unit 550 may be suitable for use astraining unit 330 (FIG. 3), although the scope of the invention is notlimited in this respect. Training unit 550 is depicted as a trainingunit that yields {tilde over (W)}(k)and μ_(t)(k) and may be suitable foruse in a two-transmitter antenna system, although the scope of theinvention is not limited in this respect. In these embodiments,spatially uncorrelated noise may be equal across all receiver antennas.In instances in which the noise is not equal across receiver antennas,one or more multipliers may be used at the output stage of Fouriertransform circuitry 154 (FIG. 1) to render equal noise equal across thereceiver antennas, causing the effective channel estimate vector to bemodified by the value of the multiplier. Training unit 550 may be usedto provide the training parameters for equalizer 302 (FIG. 3) andsoft-bit demapper 304 (FIG. 3). Because the number of transmitterantennas is fixed at two, each tone has two μ_(t)(k) values: μ₁ and μ₂,both of which are functions of tone but notation indicative of this isdropped for the sake of simplicity, μ₁ and μ₂ may comprise μ_(t)(k) andmay be provided to soft-bit demapper 304 (FIG. 3) as trainingparameters. Further w ₁ and w ₂ are row vectors, each having a dimensionof R. Jointly, w ₁ and w ₂ comprise {tilde over (W)}(k). Specifically, w₁ and w ₂ each may constitute a row in the matrix {tilde over (W)}(k),with w ₁ constituting the first row, and w ₂ constituting the secondrow. Therefore, w ₁ and w ₂ may be supplied to equalizer 302 (FIG. 3),constituting the training parameter {tilde over (W)}(k).

Training unit 550 receives three inputs: δ,h ₁, and h ₂. δ representsthe average energy of the noise (σ_(n) ²) divided by σ_(x) ². Asdiscussed above, σ_(x) ² may be defined as one, meaning that δ maysimply equal σ_(n) ² . h ₁, and h ₂ are both column vectors of dimensionR. h ₁ corresponds to the channel estimate for the first transmissionantenna, and h ₂ corresponds to the channel estimate for the secondtransmission antenna. Jointly, h ₁ and h ₂ may constitute H(k) for atwo-transmitter system. h ₁, and h ₂ are functions of tone, but notationindicative of this is dropped for the sake of simplicity.

As can be seen from FIG. 5, training unit 550 includes three innerproduct modules (generally shown as 500, 502 and 504), three conjugatetranspose modules (generally shown as 506, 508 and 510), six addermodules (generally shown as 512, 514, 516, 518, 520 and 522), twonegative reciprocating modules 524 and 526, magnitude squaring module528, and six multiplication modules (generally shown as 530, 532, 534,536, 538 and 540). Each of the aforementioned modules may be employed inhardware (e.g., as circuits within an ASIC), or may be employed insoftware or firmware, and may be stored on a medium, such as a memorydevice.

The various modules of training unit 550 operate as follows. The innerproduct modules 500-504 may each receive a first and second inputvector. Each inner product module 500-504 finds the conjugate transposeof the first input, and performs a matrix multiplication operation uponthe conjugate transpose and the second input, yielding an inner product.Conjugate transpose modules 506-510 may each operate to find theconjugate transpose of their input. Six adder modules 512-522 receivefirst and second inputs, and may yield the sum of the inputs. Twonegative reciprocating modules 524 and 526 receive a real-valued input,and yield a negative reciprocal. Magnitude squaring module 528 receivesan input, which may be a complex number, and may find the square of itsmagnitude. Finally, multiplication modules 530-540 may each receive twoinputs and yield their product.

As a consequence of the design of training unit 550, and the operationof the modules as just described, the training unit functions so as tofind the following training parameters according to the followingformulas.w ₁ =h ₁ ^(†)+(−1/( h ₂ ^(†) h ₂+δ))( h ₁ ^(†) h ₂) h ₂ ^(†)w ₂ =h ₂ ^(†)+(−1/( h ₁ ^(†) h ₁+δ))( h ₁ ^(†) h ₂)* h ₁ ^(†)μ₁ =h ₁ ^(†) h ₁+(−1/( h ₂ ^(†) h ₂+δ))| h ₁ ^(†) h ₂|²μ₂ =h ₂ ^(†) h ₂+(−1/( h ₁ ^(†) h ₁+δ))| h ₁ ^(†) h ₂|²

The above described training scheme may exhibit a substantial reductionin complexity. For example, in the context of a two-transmission antennaand four-reception antenna system, the training unit 550 may trainequalizer 302 (FIG. 3) and demapper 304 (FIG. 3) in approximately 25complex multiplications and 2 real divisions, as opposed to 158 complexmultiplications and 6 real divisions, which may be required by someconventional training techniques.

Some of the complexity reductions are owed to the adjoint of a 2×2matrix that may easily be determined without the need formultiplications. Additionally, complexity reductions are realizedbecause training unit 550 does not explicitly compute the determinant ofH^(†)(k)H(k)+δI_(T).

In training unit 550, multiplication modules 530 and 534 receive as aninput the value yielded by their respective reciprocation modules 524and 526. Thus, multiplication modules 530 and 534 operate in immediatesuccession to the operation of reciprocation modules 524 and 526.Notably, reciprocation modules 524 and 526 may take a relatively largernumber of clock cycles to operate than the other modules (multiplicationmodules, adding modules, etc.). Thus, depending upon the design ofreciprocation modules 524 and 526, multiplication modules 530 and 534may have to sit idle for one or more clock cycles, awaiting thecompletion of the operation of reciprocation modules 524 and 526. Suchan idle period may protract the training period exhibited by trainingunit 550.

FIG. 6 illustrates a training unit in accordance with some otherembodiments of the present invention. Training unit 650 may be suitablefor use as training unit 330 (FIG. 3), although the scope of theinvention is not limited in this respect. Training unit 650 addressesthe potential issue regarding protraction of the training period due topotential idle periods exhibited by multiplication modules 530 (FIG. 5)and 534 (FIG. 5). As can be seen from FIG. 6, training unit 650 includesthree inner product modules (generally shown as 600, 602 and 604), threeconjugate transpose modules (generally shown as 606, 608 and 610), sixadder modules (generally shown as 612, 614, 616, 618, 620 and 622), twonegative reciprocating modules 624 and 626, magnitude squaring module628, and six multiplication modules (generally shown as 630, 632, 634,636, 638 and 640). Each of the aforementioned modules may be employed inhardware (e.g., as circuits within an ASIC), or may be employed insoftware or firmware, and may be stored on a medium, such as a memorydevice.

As can be seen, training unit 650 is similar in structure to trainingunit 550 (FIG. 5), with some exceptions. Specifically, multiplicationmodules 538 (FIG. 5) and 540 (FIG. 5) have been relocated to receive oneof their inputs from modules 508 (FIG. 5) and 502 (FIG. 5),respectively. Thus, multiplication modules 538 (FIG. 5) and 540 (FIG. 5)are depicted as multiplication modules 630 and 634 in FIG. 6. Therearrangement of multiplication modules 538 and 540 necessitates therelocation of multiplication modules 530 (FIG. 5) and 534 (FIG. 5) to aposition immediately successive to multiplication modules 630 and 634.The effect of these two modifications is that multiplication modules 630and 634 may operate during the same period that reciprocation modules624 and 626 are operating. For example, h ₁ ^(†) h ₂ may be multipliedby h ₂ ^(†)in multiplication module 634 while reciprocation module 626operates. In training unit 550 (FIG. 5), the operation is performedafter reciprocation, not at the same time. Therefore, training unit 650may exhibit a greater degree of parallel operation than does trainingunit 550 (FIG. 5), resulting in possibly a shorter training period.

Because of these modifications, the intermediate quantities calculatedby training unit 650 may differ from those of training unit 550 (FIG.5). However, the net effect of training unit 650 is to calculatequantities identical to those of training unit 550 (FIG. 5). In otherwords, in the context of training unit 650,w ₁ =h ₁ ^(†)+(−1/( h ₂ ^(†) h ₂+δ))( h ₁ ^(†) h ₂) h ₂ ^(†)w₂ =h ₂ ^(†)+(−1/( h ₁ ^(†) h ₁+δ))( h ₁ ^(†) h ₂)* h ₁ ^(†)μ₁ =h ₁ ^(†) h ₁+(−1/( h ₂ ^(†) h ₂+δ))| h ₁ ^(†) h ₂|²μ₂ =h ₂ ^(†) h ₂+(−1/( h ₁ ^(†) h ₁+δ))| h ₁ ^(†) h ₂|².

FIG. 7 illustrates a training unit in accordance with some otherembodiments of the present invention. Training unit 750 may be suitablefor use as training unit 330 (FIG. 3), although the scope of theinvention is not limited in this respect. Training unit 750 may generate{tilde over (W)}(k) and μ_(t)(k) in the context of a three-transmitterantenna system. In these embodiments, spatially uncorrelated noise maybe equal across the receiver antennas. When the noise is not equalacross receiver antennas, one or more multipliers may be used at theoutput stage of Fourier transform circuitry 154 (FIG. 1) to render equalnoise equal across all receivers, causing the effective channel estimatevector to be modified by the value of the multiplier, although the scopeof the invention is not limited in this respect. Training unit 750 maybe used to provide the training parameters for equalizer 302 (FIG. 3)and soft-bit demapper 304 (FIG. 3). Because the number of transmitterantennas in these embodiments is fixed at three, there exist threeμ_(t)(k) values for each tone: μ₁, μ₂ and μ₃, all of which are functionsof tone, but notation indicative of this is dropped for the sake ofsimplicity. Therefore, μ₁, μ₂ and μ₃ constitute μ_(t)(k), and may beprovided to soft-bit demapper 304 (FIG. 3) as training parameters.Further, it may be understood that w ₁, w ₂ and w ₃ are row vectors,each having a dimension of R. Jointly, w ₁, w ₂ and w ₃ constitute{tilde over (W)}(k). Specifically, w ₁, w ₂ and w ₃ each constitute arow in the matrix {tilde over (W)}(k), with w ₁ constituting the firstrow, w ₂ constituting the second row, w ₃ constituting the third row.Therefore, w ₁, w ₂ and w ₃ may be supplied to equalizer 302 (FIG. 3),constituting the training parameter {tilde over (W)}(k).

Training unit 750 receives four inputs: δ, h ₁, h₂ and h₃ . δ representsthe average energy of the noise (σ_(n) ²) divided by σ_(x) ². Asdiscussed above, σ_(x) ² may be defined as one, meaning that δ maysimply equal σ_(n) ². h ₁, h ₂ and h ₃ are column vectors of dimensionR. h ₁ corresponds to the channel estimate for the first transmissionantenna, h ₂ corresponds to the channel estimate for the secondtransmission antenna, and h ₃ corresponds to the channel estimate forthe third transmission antenna. Jointly, h ₁, h ₂ and h ₃ may constituteH(k) for a three-transmitter system. h ₁, h ₂ and h ₃ are functions oftone, but notation indicative of this is dropped for the sake ofsimplicity. For the sake of simple illustration, h ₁, h ₂, and h ₃ areeach depicted as inputs at three locations in FIG. 7. Each instance of h₁ refers to the same column vector, each instance h ₂ refers to the samecolumn vector, and each instance of h ₃ refers to the same columnvector.

As can be seen from FIG. 7, training unit 750 includes six inner productmodules (generally shown as 700, 702, 704, 706, 708 and 710), adjointcalculation module 712, determinant calculation module 714, threereciprocation units 716 (designated with “1/”), fifteen multiplier units(designated with “x”), six adder units (designated with “+”), threedifference units 718, and three conjugate transpose modules (designatedwith “†”). Although FIG. 7 depicts a particular number of modules andunits, training unit 750 may include additional or fewer modules andunits. Each of the aforementioned modules may be employed in hardware(as circuits within an ASIC), or may be employed in software orfirmware, and may be stored on a medium, such as a memory device.

The various modules of training unit 750 operate as follows. Innerproduct modules 700-710 each receive a first and second input vector.Each inner product module 700-710 finds the conjugate transpose of thefirst input (depicted as the upper input of training unit 750), andperforms a matrix multiplication operation upon the conjugate transposeand the second input (depicted as the lower input of training unit 750),yielding an inner product. The conjugate transpose modules each operateto find the conjugate transpose of their input. The six adder modulesreceive first and second inputs, and yield the sum of the inputs. Thethree reciprocating modules 716 receive an input, and yield itsreciprocal. The fifteen multiplication modules each receive two inputs,and yield their product. The three difference modules 718 receive firstand second inputs, and yield the difference of the inputs.

The output of each inner product modules 700-710 may be termed γ₁-γ₆,respectively. Thus the output of inner product module 700 h ^(†) ₁ h ₁,which is termed γ₁. Similarly, the output of inner product module 702 ish ^(†) ₂ h ₂, which is termed γ₂, and so on. Jointly, inner productmodules 700-710 construct quantities sufficient to define the followingmatrix, noting that δ is an input value of training unit 750:$\quad\begin{bmatrix}{\gamma_{1} + \delta} & \gamma_{4} & \gamma_{5} \\\gamma_{4}^{*} & {\gamma_{2} + \delta} & \gamma_{6} \\\gamma_{5}^{*} & \gamma_{6}^{*} & {\gamma_{3} + \delta}\end{bmatrix}$

The above matrix may be referred to as B for ease of reference. Adjointcalculation module 712 may calculate the adjoint of matrix B. Suchcalculation is known and is therefore not discussed herein. Adjointcalculation module 712 may include circuits for multiplying, adding, andsubtracting, in order to carry out the adjoint calculation.

In some embodiments, adjoint calculation module 712 generates ninevalues, g₁, g₂, g₃, g₄, g₅, g₆, g₄*, g₅*, and g₆*, which constitute amatrix, G, which is the adjoint of B: $G = \begin{bmatrix}g_{1} & g_{4} & g_{5} \\g_{4}^{*} & g_{2} & g_{6} \\g_{5}^{*} & g_{6}^{*} & g_{3}\end{bmatrix}$

Determinant calculation module 714 calculates the determinant of B,which is termed Δ herein. Determinant calculation module 714 may includecircuits for multiplying, adding, and subtracting, in order to carry outthe determinant calculation.

Based on the operations described above, training unit 750 may determinethe following training parameters according to the following formulas.w ₁ =h ₁ ^(†)+(g ₄ /g ₁) h ₂ ^(†)+(g ₅ /g ₁) h ₃ ^(†)w ₂ =h ₂ ^(†)+(g ₄ */g ₂) h ₁ ^(†)+(g ₆ /g ₂) h ₃ ^(†)w ₃ =h ₃ ^(†)+(g ₅ */g ₃) h ₁ ^(†)+(g ₆ */g ₃) h ₂ ^(†)μ₁=Δ(1/g ₁)−δμ₂=Δ(1/g ₂)−δμ₃=Δ(1/g ₃)−δ

The above described training scheme may exhibit a significationreduction in complexity, although the scope of the invention is notlimited in this respect. For example, in some embodiments in the contextof a three-transmission antenna and four-reception antenna system,training unit 750 may train equalizer 302 (FIG. 3) and demapper 304(FIG. 3) in, for example, approximately 72 complex multiplications andapproximately 3 real divisions, as opposed to approximately 220 complexmultiplications and approximately 7 real divisions required by someconventional training schemes.

Complexity reductions are owed principally to the fact that the adjointof a 3×3 matrix may be easily determined by finding the cofactor of eachelement in the 3×3 matrix, and then transposing the matrix, none ofwhich requires a division operation.

To achieve reductions in complexity, in general, it is useful to have ascheme by which a linear matrix equation can be solved withoutemployment of a division operation. For example, in the case ofcalculating MMSE equalizer matrix (e.g., for training of an MMSEequalizer, such as equalizer 202 (FIG. 2)),W ^(†) _(MMSE)(k)=(H(k)H ^(†)(k)+(R _(nn)(k)/σ_(x) ²))⁻¹ ·H(k),

which is an example of a linear matrix equation. Similarly, in the caseof calculating a zero force (ZF) equalizer matrix for training of a ZFequalizer,W ^(†) _(ZF)(k)=(H ^(†)(k)H(k))⁻¹ ·H ^(†)(k),

which is another example of a linear matrix equation.

For either form of equalizer matrix, it is advantageous to solve thelinear matrix equations without use of a division operation, becausedivision operations generally take a relatively longer period forexecution than do other operations. To accomplish this result, a scaledversion of the equalizer matrix may be solved for, as long as otherelements of training unit are provided with the scaling factor, as shownand discussed above with reference to FIG. 4.

Both equalizer equations may be generally rewritten as the followingform of the following linear matrix equation:Ψ·W=θ,

where Ψ represents a known matrix, such as H^(†)(k)H(k), W represents anunknown matrix, such as an equalizer matrix that is to be calculated,and θ represents another known matrix, such as H(k) (or, in the instanceof attempting to find an inverse matrix, I_(N)).

For the sake of generality, ${\Psi = \begin{bmatrix}A & B \\C & D\end{bmatrix}},{W\quad = \begin{bmatrix}P & Q \\R & S\end{bmatrix}},{{{and}\quad\theta}\quad = \quad\begin{bmatrix}T & V \\X & Y\end{bmatrix}},$

where A, B, C, D, P, Q, R, S, T, V, X, and Y are matrices. In otherwords, Ψ, W, and θ are represented as matrices of matrices. Therefore,the problem may be represented as: ${\begin{bmatrix}A & B \\C & D\end{bmatrix} \cdot \begin{bmatrix}P & Q \\R & S\end{bmatrix}} = \begin{bmatrix}T & V \\X & Y\end{bmatrix}$

By virtue of the above-shown expression of the problem of generallysolving a linear matrix equation, the relationship between A, B, C, D,P, Q, R, S, T, V, X, and Y is:A·P+B·R=T  Equation (1)A·Q+B·S=V  Equation (2)C·P+D·R=X  Equation (3)C·Q+D·S=Y  Equation (4)

FIG. 8 is a flow chart of a matrix processing procedure in accordancewith some embodiments of the present invention. Matrix processingprocedure 800 may be used to find scaled versions of matrices P, Q, R,and S (denoted P_(s), Q_(s), R_(s), and S_(s)), although otherprocedures may also be used. Operations 802 through 818 of matrixprocessing procedure 800 illustrate a procedure for finding the scalefactor and matrices P_(s) and R_(s) and involve manipulation of theEquations (1) and (3) within quadripartite relationship of matrices A,B, C, D, P, Q, R, S, T, V, X, and Y. Operation 820 of matrix processingprocedure 800 may be used to find matrices Q_(s) and S_(s), and, in thatcase, involves manipulation of Equations (2) and (4).

Although the description of matrix processing procedure 800 is describedas a sequence of mathematical steps, it is understood that a unit ofsoftware, or a set of hardware modules may implement matrix processingprocedure 800, and that the subjects of manipulation may comprise datavalues represented by matrices A, B, C, D, P, Q, R, S, T. V, X, and Y.In the context of finding an equalizer matrix, the values beingmanipulated may include elements within a channel estimate matrix,elements within a conjugate transpose of the channel estimate matrix,and elements making up the noise correlation matrix. By suchmanipulation, matrices P_(s), Q_(s), R_(s), and S_(s) may be foundwithout use of a division operation, meaning that an equalizer may beefficiently trained by such a scheme.

Operation 802 comprises computing the determinant and the adjoint ofmatrix A.

Operation 804 comprises computing matrix J by pre-multiplying matrix Bwith the adjoint of matrix A, and computing matrix K by pre-multiplyingmatrix T with the adjoint of matrix A.

Operation 806 comprises computing matrix E by subtracting the result ofthe pre-multiplication of matrix J by matrix C, from the result of themultiplication of matrix D by the determinant of matrix A.

Operation 808 comprises computing the determinant and the adjoint ofmatrix E.

Operation 810 comprises computing matrix F by subtracting the result ofthe pre-multiplication of matrix K by matrix C, from the result of themultiplication of matrix X by the determinant of matrix A.

Operation 812 comprises computing matrix R₁ by pre-multiplying matrix Fwith the adjoint of matrix E.

Operation 814 comprises computing matrix P_(s) by subtracting the resultof the pre-multiplication of matrix R₁ by matrix J, from the result ofthe multiplication of matrix K by the determinant of matrix E.

Operation 816 comprises computing matrix R_(s) by multiplying matrix R₁by the determinant of matrix A.

Operation 818 comprises computing scale factor s by multiplying thedeterminant of matrix A by the determinant of matrix E.

Upon the completion of operations 802 through 818, matrices P_(s) andR_(s) are determined in terms of matrices A, B, C, D, T and X. Operation820 comprises repeating operations 802 through 816 to determine matricesQ_(s) and S_(s) in terms of matrices A, B, C, D, V and Y. Operation 820may include substituting matrix Q_(s) for matrix P_(s), matrix S_(s) formatrix R_(s), matrix V for matrix T, and matrix Y for matrix X. Upon thecompletion of procedure 800, the scale factor s and the scaled versionsof P, Q, R and S (denoted as P_(s), Q_(s), R_(s), and S_(s)) are knownin terms of known matrices A, B, C, D, T, V, X and Y.

In some embodiments, P_(s), Q_(s), R_(s), and S_(s), may be used as theequalizer coefficient matrices generated by a training unit, such astraining unit 158 (FIG. 1), and/or training unit 430 (FIG. 4), althoughthe scope of the invention is not limited in this respect.

The following description may illustrate the derivation of procedure.Initially, Equation (1) is manipulated to find P in terms of a partialscaling factor, the determinant of A:det(A)·P=−J·R+K,  Equation (5)

where J=adj(A)·B and K=adj(A)·T. Thus, the determinant of A is the firstpartial scaling factor.

Next, Equation (3) is pre-multiplied by the first partial scaling factor(i.e., the determinant of A), and is simplified, to arrive at anequation for R in terms of a second partial scaling factor:det(E)·R=adj(E)·F,  Equation (6)

where E=det(A)·D−C·J, and F=−C·K+det(A)·X. Thus, the determinant of E isthe second partial scaling factor.

Next, Equation (5) is multiplied by the second partial scaling factor(i.e., the determinant of E) using Equation (6) to arrive at theequation for the scaled version of P:P _(s) =−J·adj (E)·F+det(E)·K,  Equation (7)

where the scaling factor, s, can be determined by Equation (8) below:s=det(A)·det(e).  Equation (8)

Next, Equation (6) is multiplied by the first partial scaling factor(i.e., the determinant of A) to arrive at the equation for the scaledversion of R:R _(s)=det(A)·adj(e)·F.  Equation (9)

Broadly speaking, matrix processing procedure 800 solves for scaledversion of unknowns and the scaling factor. When the solving processcalls for calculation of an inverse matrix, the adjoint may be usedinstead, while the determinant may be identified as a partial scalingfactor. At the end, all of the partial scaling factors (i.e.,determinants) may be multiplied together to arrive at a final scalingfactor.

Using matrix processing procedure 800 to determine Q_(s), and S_(s), oneobtains:Q _(s) =−J·adj(E)·G+det(E)·LS _(s)=det(A)·adj(E)·G,where L=adj(A)·V, and G=−C·L+det(A)·Y.

Although the individual operations of procedure 800 are illustrated anddescribed as separate operations, one or more of the individualoperations may be performed concurrently, and nothing requires that theoperations be performed in the order illustrated.

Unless specifically stated otherwise, terms such as processing,computing, calculating, determining, displaying, or the like, may referto an action and/or process of one or more processing or computingsystems or similar devices that may manipulate and transform datarepresented as physical (e.g., electronic) quantities within aprocessing system's registers and memory into other data similarlyrepresented as physical quantities within the processing system'sregisters or memories, or other such information storage, transmissionor display devices. Furthermore, as used herein, computing deviceincludes one or more processing elements coupled with computer-readablememory that may be volatile or non-volatile memory or a combinationthereof.

Some embodiments of the invention may be implemented in one or acombination of hardware, firmware and software. Embodiments of theinvention may also be implemented as instructions stored on amachine-readable medium, which may be read and executed by at least oneprocessor to perform the operations described herein. A machine-readablemedium may include any mechanism for storing or transmitting informationin a form readable by a machine (e.g., a computer). For example, amachine-readable medium may include read-only memory (ROM),random-access memory (RAM), magnetic disk storage media, optical storagemedia, flash-memory devices, electrical, optical, acoustical or otherform of propagated signals (e.g., carrier waves, infrared signals,digital signals, etc.), and others.

The Abstract is provided to comply with 37 C.F.R. Section 1.72(b)requiring an abstract that will allow the reader to ascertain the natureand gist of the technical disclosure. It is submitted with theunderstanding that it will not be used to limit or interpret the scopeor meaning of the claims.

In the foregoing detailed description, various features are occasionallygrouped together in a single embodiment for the purpose of streamliningthe disclosure. This method of disclosure is not to be interpreted asreflecting an intention that the claimed embodiments of the subjectmatter require more features than are expressly recited in each claim.Rather, as the following claims reflect, invention may lie in less thanall features of a single disclosed embodiment. Thus the following claimsare hereby incorporated into the detailed description, with each claimstanding on its own as a separate preferred embodiment.

1. A multicarrier receiver comprising: an equalizer to multiplyfrequency domain signals generated from each of a plurality of receivesignal paths by an equalizer coefficient matrix to remove at least somecrosstalk and noise to generate spatially equalized frequency domainsignals for each subcarrier; a soft-bit demapper for each spatialchannel to multiply the spatially equalized frequency domain signals bysoft-bit weights for use in generating soft bits; and a training unit togenerate the equalizer coefficient matrix for each subcarrier and togenerate the soft-bit weights from effective channel taps.
 2. Thereceiver of claim 1 wherein the receiver receives two or more spatialdata streams concurrently through two or more receive antennas, the twoor more spatial data streams having been transmitted by a correspondingtwo or more transmit antennas of a transmitting station.
 3. The receiverof claim 2 wherein the effective channel taps represent diagonalelements of an effective channel matrix, and wherein the training unitgenerates the effective channel matrix by multiplying the equalizercoefficient matrix by a channel estimate matrix.
 4. The receiver ofclaim 3 wherein the multiplication of the spatially equalized frequencydomain signals by the soft-bit weights generates soft-bit weightedspatially equalized frequency domain signals, and wherein the soft-bitdemapper includes a soft-bit generator to generate the soft bits fromthe soft-bit weighted spatially equalized frequency domain signals,values proportional to the soft-bit weights, and a square root oftransmit sub-symbol energy, and wherein a number of soft bits generatedby the soft-bit demapper depends on a modulation level used by atransmitting station.
 5. The receiver of claim 2 wherein the trainingunit generates the equalizer coefficient matrix from a noise correlationmatrix, a channel estimate matrix and an average energy of transmittedsub-symbols.
 6. The receiver of claim 2 wherein the training unitgenerates the equalizer coefficient matrix for each subcarrier of areceived multicarrier signal for each received packet from traininginformation in each received packet.
 7. The receiver of claim 1 furthercomprising: Fourier transform circuitry for each receive antenna togenerate the frequency domain signals for each receive signal path; aspace-frequency de-interleaver comprising deinterleavers and a spatialde-commutator, the deinterleavers to perform a deinterleaving operationon the soft bits generated from soft-bit demappers for each spatialchannel, the spatial de-commutator to combine the deinterleaved softbits generated from the deinterleavers; and a decoder to recoverhard-bits and generate a decoded bit stream from the combineddeinterleaved soft bits provided by the spatial de-commutator.
 8. Thereceiver of claim 7 wherein the decoded bit stream corresponds to aplurality of spatial data streams transmitted by a plurality of transmitantennas of a transmitting station, and wherein each transmit antennatransmits one of the spatial data streams over a corresponding spatialchannel.
 9. The receiver of claim 7 further comprising a plurality ofmultipliers to multiply the frequency domain signals provided by theFourier transform circuitry for each receive signal path to helpequalize noise power across the receive antennas when the noise power isnot substantially equal across the receive antennas.
 10. The receiver ofclaim 1 wherein the soft bits comprise real values representing aconfidence level for a corresponding hard-bit.
 11. The receiver of claim1 wherein the training unit generates the soft-bit weights from achannel estimate matrix, an average energy of transmitted sub-symbols,and a matrix related to a noise correlation matrix.
 12. The receiver ofclaim 1 wherein the training unit calculates a set of operationalparameters for the equalizer, and wherein the training unit calculates aset of operational parameters for the soft-bit demappers based ondiagonal elements of an effective channel matrix without the use ofnon-diagonal elements in the effective channel matrix.
 13. The receiverof claim 12 wherein when calculating the set of operational parameters,the training unit finds a set of reciprocals, each member of the setbeing a reciprocal of the difference between one and each diagonal entryin the effective channel matrix, and wherein when calculating the set ofoperational parameters, the training unit further subtracts a value ofone from each reciprocal in the set to yield a set of operationalparameters for the soft-bit demappers.
 14. The receiver of claim 12wherein the training unit: calculates a matrix C(k) by premultiplying aninverse of a normalized noise correlation matrix D(k) by the conjugatetranspose of the channel estimate matrix; calculates a matrix B(k) byadding an identity matrix to the result of the premultiplying of thechannel estimate matrix by the matrix C(k); computes a matrix G(k) byinverting the matrix B(k); and computes the soft-bit weights byreciprocating corresponding diagonal elements of the matrix G(k),wherein the identity matrix has a dimension equal to a number of thetransmit antennas, wherein a diagonal soft-bit weight matrix haselements corresponding to the soft-bit weights, and wherein theequalizer coefficient matrix is computed by premultiplying the result,of premultiplying matrix C(k) by matrix G(k), by the diagonal soft-bitweight matrix.
 15. The receiver of claim 5 wherein the equalizercoefficient matrix is a scaled version of the equalizer coefficientmatrix, wherein the scaled version of the equalizer coefficient matrixis used to compute effective channel taps, wherein the soft-bit weightsare computed by reciprocating the result of subtracting thecorresponding effective channel taps from the scale factor, and whereinthe operational parameters for the soft-bit demappers are calculated bysubtracting one from the result of multiplication of the soft-bitweights by the scale factor.
 16. The receiver of claim 2 wherein thetransmitting station transmits with two transmit antennas and thereceiver receives with at least two receive antennas, wherein thetraining unit comprises a plurality of inner product modules, aplurality of conjugate transpose modules, a plurality of adder modules,a plurality of negative reciprocating modules, a magnitude squaringmodule and a plurality of multiplication modules, wherein each the innerproduct module receives first and second vector inputs, computes aconjugate transpose of the first vector input, and performs a matrixmultiplication on the conjugate transpose and the second vector input toyield an inner product.
 17. The receiver of claim 2 wherein thetransmitting station transmits with two transmit antennas and thereceiver receives with at least two receive antennas, wherein thetraining unit comprises a plurality of inner product modules, aplurality of conjugate transpose modules, a plurality of adder modules,a plurality of negative reciprocating modules, a magnitude squaringmodule, and a plurality of multiplication modules, wherein themultiplication modules and the reciprocating units are arranged tooperate substantially concurrently.
 18. The receiver of claim 2 whereinthe transmitting station transmits with three transmit antennas and thereceiver receives with at least three receive antennas, wherein thetraining unit comprises a plurality of inner product modules, an adjointcalculation module, a determinant calculation module, a plurality ofreciprocation units, a plurality of multiplier units, a plurality ofadder units, a plurality of difference units, and three conjugatetranspose modules.
 19. A method for generating soft bits within amulticarrier receiver comprising: multiplying frequency domain signalsgenerated from each of a plurality of receive signal paths by anequalizer coefficient matrix to remove at least some crosstalk and noiseto generate spatially equalized frequency domain signals for eachsubcarrier; multiplying the spatially equalized frequency domain signalsby soft-bit weights for use in generating soft bits; and generating theequalizer coefficient matrix for each subcarrier and the soft-bitweights from effective channel taps.
 20. The method of claim 19 furthercomprising receiving two or more spatial data streams concurrentlythrough two or more receive antennas, the two or more spatial datastreams having been transmitted by a corresponding two or more transmitantennas of a transmitting station, and wherein the soft bits correspondto each of the two or more spatial data streams.
 21. The method of claim20 wherein the effective channel taps represent diagonal elements of aneffective channel matrix, and wherein the method further comprisesgenerating the effective channel matrix by multiplying the equalizercoefficient matrix by a channel estimate matrix.
 22. The method of claim21 wherein the multiplication of the spatially equalized frequencydomain signals by the soft-bit weights generates soft-bit weightedspatially equalized frequency domain signals, and wherein the methodfurther comprises generating the soft bits from the soft-bit weightedspatially equalized frequency domain signals, values proportional to thesoft-bit weights, and a square root of transmit sub-symbol energy, andwherein a number of soft bits generated depends on a modulation levelused by a transmitting station.
 23. The method of claim 20 furthercomprising generating the equalizer coefficient matrix from a noisecorrelation matrix, a channel estimate matrix and an average energy oftransmitted sub-symbols.
 24. The method of claim 20 further comprisinggenerating the equalizer coefficient matrix for each subcarrier of areceived multicarrier signal for each received packet from traininginformation in each received packet.
 25. The method of claim 19 furthercomprising: generating the frequency domain signals for each receivesignal path; performing a deinterleaving operation on the soft bitsgenerated for each spatial channel; combining the deinterleaved softbits; and recovering hard-bits and generating a decoded bit stream fromthe combined deinterleaved soft bits.
 26. The method of claim 25 whereinthe decoded bit stream corresponds to a plurality of spatial datastreams transmitted by a plurality of transmit antennas of atransmitting station, each transmit antenna having transmitted one ofthe spatial data streams over a corresponding spatial channel.
 27. Themethod of claim 25 further comprising multiplying the frequency domainsignals for each receive signal path to help equalize noise power acrossthe receive antennas when the noise power is not substantially equalacross the receive antennas.
 28. The method of claim 19 wherein the softbits comprise real values representing a confidence level for acorresponding hard-bit.
 29. The method of claim 19 wherein the soft-bitweights are generated from a channel estimate matrix, an average energyof transmitted sub-symbols, and a matrix related to a noise correlationmatrix.
 30. The method of claim 19 further comprising generating a setof operational parameters for performing an equalization, and generatinga set of operational parameters for soft-bit demapping based on diagonalelements of an effective channel matrix without the use of non-diagonalelements in the effective channel matrix.
 31. The method of claim 30wherein calculating the set of operational parameters includes finding aset of reciprocals, each member of the set being a reciprocal of thedifference between one and each diagonal entry in the effective channelmatrix, and wherein calculating the set of operational parametersfurther includes subtracting a value of one from each reciprocal in theset to yield a set of operational parameters for the soft-bit demappers.32. The method of claim 30 wherein the training unit: calculates amatrix C(k) by premultiplying an inverse of a normalized noisecorrelation matrix D(k) by the conjugate transpose of the channelestimate matrix; calculates a matrix B(k) by adding an identity matrixto the result of the premultiplying of the channel estimate matrix bythe matrix C(k); computes a matrix G(k) by inverting the matrix B(k);and computes the soft-bit weights by reciprocating correspondingdiagonal elements of the matrix G(k), wherein the identity matrix has adimension equal to a number of the transmit antennas wherein a diagonalsoft-bit weight matrix has elements corresponding to the soft-bitweights, and wherein the equalizer coefficient matrix is computed bypremultiplying the result, of premultiplying matrix C(k) by matrix G(k),by the diagonal soft-bit weight matrix.
 33. The method of claim 23wherein the equalizer coefficient matrix is a scaled version of theequalizer coefficient matrix, wherein the scaled version of theequalizer coefficient matrix is used to compute effective channel taps,wherein the soft-bit weights are computed by reciprocating the result ofsubtracting the corresponding effective channel taps from the scalefactor, and wherein the operational parameters for the soft-bitdemappers are calculated by subtracting one from the result ofmultiplication of the soft-bit weights by the scale factor.
 34. Themethod of claim 20 wherein the transmitting station transmits with twotransmit antennas and the receiver receives with at least two receiveantennas, wherein the training unit comprises a plurality of innerproduct modules, a plurality of conjugate transpose modules, a pluralityof adder modules, a plurality of negative reciprocating modules, amagnitude squaring module and a plurality of multiplication modules,wherein each the inner product module receives first and second vectorinputs, computes a conjugate transpose of the first vector input, andperforms a matrix multiplication on the conjugate transpose and thesecond vector input to yield an inner product.
 35. The method of claim20 wherein the transmitting station transmits with two transmit antennasand the receiver receives with at least two receive antennas, whereinthe training unit comprises a plurality of inner product modules, aplurality of conjugate transpose modules, a plurality of adder modules,a plurality of negative reciprocating modules, a magnitude squaringmodule, and a plurality of multiplication modules, wherein themultiplication modules and the reciprocating units are arranged tooperate substantially concurrently.
 36. The method of claim 20 whereinthe transmitting station transmits with three transmit antennas and thereceiver receives with at least three receive antennas, wherein thetraining unit comprises a plurality of inner product modules, an adjointcalculation module, a determinant calculation module, a plurality ofreciprocation units, a plurality of multiplier units, a plurality ofadder units, a plurality of difference units, and three conjugatetranspose modules.
 37. A system comprising a multicarrier receiver; andtwo or more substantially omnidirectional antennas coupled to thereceiver, each antenna associated with one of a plurality of receivesignal paths, the receiver comprising: an equalizer to multiplyfrequency domain signals generated from each of the receive signal pathsby an equalizer coefficient matrix to remove at least some crosstalk andnoise to generate spatially equalized frequency domain signals for eachsubcarrier; a soft-bit demapper for each spatial channel to multiply thespatially equalized frequency domain signals by soft-bit weights for usein generating soft bits; and a training unit to generate the equalizercoefficient matrix for each subcarrier and to generate the soft-bitweights from effective channel taps.
 38. The system of claim 37 whereinthe receiver receives two or more spatial data streams concurrentlythrough two or more receive antennas, the two or more spatial datastreams having been transmitted by a corresponding two or more transmitantennas of a transmitting station.
 39. The system of claim 38 whereinthe effective channel taps represent diagonal elements of an effectivechannel matrix, and wherein the training unit generates the effectivechannel matrix by multiplying the equalizer coefficient matrix by achannel estimate matrix.
 40. The system of claim 39 wherein themultiplication of the spatially equalized frequency domain signals bythe soft-bit weights generates soft-bit weighted spatially equalizedfrequency domain signals, and wherein the soft-bit demapper includes asoft-bit generator to generate the soft bits from the soft-bit weightedspatially equalized frequency domain signals, values proportional to thesoft-bit weights, and a square root of transmit sub-symbol energy, andwherein a number of soft bits generated by the soft-bit demapper dependson a modulation level used by a transmitting station.
 41. Amachine-accessible medium that provides instructions, which whenaccessed, cause a machine to perform operations comprising: multiplyingfrequency domain signals generated from each of a plurality of receivesignal paths by an equalizer coefficient matrix to remove at least somecrosstalk and noise to generate spatially equalized frequency domainsignals for each subcarrier; multiplying the spatially equalizedfrequency domain signals by soft-bit weights for use in generating softbits; and generating the equalizer coefficient matrix for eachsubcarrier and the soft-bit weights from effective channel taps.
 42. Themachine-accessible medium of claim 41 wherein the instructions, whenfurther accessed cause the machine to receive two or more spatial datastreams concurrently through two or more receive antennas, the two ormore spatial data streams having been transmitted by a corresponding twoor more transmit antennas of a transmitting station, and wherein thesoft bits correspond to each of the two or more spatial data streams.43. The machine-accessible medium of claim 42 wherein the effectivechannel taps represent diagonal elements of an effective channel matrix,and wherein the instructions, when further accessed cause the machine togenerate the effective channel matrix by multiplying the equalizercoefficient matrix by a channel estimate matrix.
 44. Themachine-accessible medium of claim 43 wherein the multiplication of thespatially equalized frequency domain signals by the soft-bit weightsgenerates soft-bit weighted spatially equalized frequency domainsignals, and wherein the instructions, when further accessed cause themachine to generate the soft bits from the soft-bit weighted spatiallyequalized frequency domain signals, values proportional to the soft-bitweights, and a square root of transmit sub-symbol energy, and wherein anumber of soft bits generated depends on a modulation level used by atransmitting station.
 45. A method for determining scaled versions offour unknown matrices P, Q, R and S in the equation ${\begin{bmatrix}A & B \\C & D\end{bmatrix} \cdot \begin{bmatrix}P & Q \\R & S\end{bmatrix}} = \begin{bmatrix}T & V \\X & Y\end{bmatrix}$ wherein matrices A, B, C, D, T, V, X and Y are known, andwherein matrices P_(s), Q_(s), R_(s), and S_(s) denote scaled versionsof matrices P, Q, R and S respectively, the method comprising: computingthe determinant and the adjoint of the matrix A; computing a matrix J bypre-multiplying the matrix B with the adjoint of the matrix A; computinga matrix K by pre-multiplying the matrix T with the adjoint of thematrix A; computing a matrix E by subtracting the result of thepre-multiplication of the matrix J by the matrix C from the result ofthe multiplication of the matrix D by the determinant of the matrix A;computing the determinant and the adjoint of the matrix E; computing amatrix F by subtracting the result of the pre-multiplication of thematrix K by the matrix C from the result of the multiplication of thematrix X by the determinant of the matrix A; computing a matrix R₁ bypre-multiplying the matrix F with the adjoint of the matrix E; computingthe matrix P_(s) by subtracting the result of the pre-multiplication ofthe matrix R₁ by the matrix J from the result of the multiplication ofthe matrix K by the determinant of the matrix E; computing the matrixR_(s) by multiplying the matrix R₁ by the determinant of the matrix A;and computing a scale factor s by multiplying the determinant of thematrix A by the determinant of the matrix E.
 46. The method of claim 45further comprising, prior to the computing steps, partitioning eachmatrix in a linear matrix equation into four sub-matrices, wherein oneknown set of the sub-matrices comprises the matrices A, B, C and D,wherein another known set of the sub-matrices comprises the matrices T,V, X and Y, and wherein an unknown set of the sub-matrices comprises thematrices P, Q, R and S.
 47. The method of claim 46 further comprisingrepeating the computing steps to determine the matrices Q_(s) and S_(s)in terms of matrices A, B, C, D, V and Y, and wherein the matricesP_(s), Q_(s), R_(s), and S_(s) are scaled by the scale factor s withrespect to matrices P, Q, R and S respectively.
 48. The method of claim47 wherein the repeating comprises substituting matrix Q_(s) for matrixP_(s), matrix S_(s) for matrix R_(s), matrix V for matrix T and matrix Yfor matrix X.
 49. The method of claim 47 wherein matrices P_(s), Q_(s),R_(s), and S_(s) correspond to equalizer coefficient matrices, whereinthe computing steps are performed by processing circuitry of amulticarrier receiver, and wherein the method further comprisingapplying the equalizer coefficient matrices to received frequency domainsignals in the multicarrier receiver of a 4×4 multiple-input,multiple-output (MIMO) communication system.